Graphs are not universal for online computability

Abstract

We show that structures with only one binary function symbol are universal for “online” (punctual) computable structures. In contrast, we give a description of punctually categorical graphs which implies that graphs are not universal for online computability.

Introduction

The study of general computable processes in algebra is a long tradition going back to the early twentieth century; see van der Waarden [45], Hermann [18], and Dehn [10]. Maltsev [35] and Rabin [39] proposed the following standard model for such investigations: A computable presentation of a structure is a coding of the structure with universe $\mathbb{N}$, so that the relations and functions are Turing computable on the codes of elements. For a group this is the same as to say that the group has a recursive presentation with solvable word problem [19], [33]. The general area of computable structures now has a well-developed theory. This area has several threads, as can be seen by the large volumes [13], [14]. Such investigations rely on the most general notion of an algorithm that we know today, namely Turing computable functions. In particular, we do not assume any time or space bounds in such abstract computations. It is a general phenomenon that many such abstract algorithms can be provably turned into more feasible algorithms for reasons yet to be understood; see, e.g., [17], [27]. The present article contributes to the research program which, among other goals, aims to explain this phenomenon.

The other important goal is to give a foundation for online structure theory. There are a large number of natural algorithmic processes which receive input over time but need to provide output promptly. After decades of development, computability theory and computable structure theory give a well-developed framework to investigate the limits of offline computation in infinite mathematics. Nonetheless, there is no such general theory for online potentially infinite structures. Following [27], we choose primitive recursion as the basis of our theory. We will briefly discuss the reasons for this, and then give some precise definitions.

What happens when we put resource bounds on the definitions of allowable computation? Authors have looked at various forms of feasible structure theory in the literature.

Khoussainov and Nerode [28] initiated a systematic study into automatically presentable algebraic structures. Automatic structures are linear-time computable and have decidable theories, but such presentations seem quite rare. For example, the additive group of the rationals is not automatic [44]. The approach via finite automata is highly sensitive to how we define what we mean by automatic. For example, treating a function as a relation yields quite a different kind of automatic presentation from treating it is as a transducer. See [11] for an alternate approach to automatic groups. Although the theory of automatic structures is a beautiful subject, a finite automaton is definitely not general enough for our purposes.

Cenzer and Remmel, Gregorieff, Alaev, and others [9], [17], [1], [2] studied polynomial time presentable structures. We omit the formal definitions, but we note that they are sensitive to how exactly we code the domain. In many common algebraic classes we can show that all Turing computable structures have polynomial-time computable copies. One attractive result is that every computably presentable linear ordering has a copy in linear time and logarithmic space [17]. Similar results hold for broad subclasses of Boolean algebras [6] and commutative groups [8], [5], and some other structures [7]. As was noted in [27], many known proofs from polynomial time structure theory (e.g., [7], [8], [5], [17]) are focused on making the operations and relations on the structure merely primitive recursive, and then observing that the presentation that we obtain is in fact polynomial-time. On the other hand, to illustrate that a structure has no polynomial time copy, it is sometimes easiest to argue that it does not even have a copy with primitive recursive operations; see, e.g., [8].

Beginning in the 1980's there has been quite a lot of work on online infinite combinatorics, particularly by Kierstead, Trotter, Remmel and others ([23], [24], [29], [34], [41]). The central definition of this framework requires an “algorithm” on a potentially infinite input to be a total function; note the function does not even have to be Turing computable. Even though this definition seems rather crude, some results were quite surprising and had depth. Work here is ongoing; see [25] for a somewhat dated survey. One of the standard references for a more practice-related theory of online computation in computer science is book [4]. Although Borodin and El-Yaniv [4] note that a practical algorithm is expected to be efficient, most (if not all) of their results hold for total not necessarily computable functions. From the more practical point of view assuming that our algorithm is at least primitive recursive is of course not much of a restriction. This assumption is essentially equivalent to saying that our algorithm belongs to some reasonable complexity class without necessarily specifying what exactly the class is. Note also that every primitive recursive function is of course total.

Because of all of the reasons outlined above, we believe that primitive recursive infinite structures provide a general model of the general issues faced in the theory of online structures. They give a generic model to focus on the key ingredient, that unbounded search is outlawed and that the algorithm must be total. It is therefore natural to systematically investigate those infinite structures that admit a presentation with primitive recursive operations, as defined below:

Definition 1.1 [27]

A countable structure is fully primitive recursive (fpr) if its domain is $\mathbb{N}$ and the operations and predicates of the structure are (uniformly) primitive recursive.

We call fpr structures computable without delay, punctually computable, or simply punctual. Here “delay” really means an instance of a truly unbounded search. We could also agree that all finite structures are also punctual by allowing initial segments of $\mathbb{N}$ to serve as their domains. Although the definition above is not restricted to finite languages, we will never consider infinite languages in the paper; therefore, we do not clarify what uniformity means in Definition 1.1.

The notion of a punctual structure leads to a rather rich theory; see [27], [36], [3], [26], [37]. Irrelevant counting combinatorics is stripped from proofs in the theory, thus emphasising the effects related to the existence of a computational bound in principle. These effects are far more significant than it may seem at first glance; the main result of the paper will be a good illustration of this phenomenon.

It is well-known that any countable algebraic structure can be algorithmically transformed into a graph. One represents elements of the structure as particular nodes in the graph, each of which is marked with a particular configuration of nodes to distinguish them from the other nodes, and then connects these particular nodes with other configurations to encode the operations and relations. This is simple enough to be considered folklore, but see [43], [32] for the first explicit use of such transformations. Such a coding preserves most of the (Turing) computability-theoretic properties of importance. Classes of countable structures which can effectively “code” any other countable structure are called universal. Examples of universal classes also include groups, partial orders, integral domains [20], and, most notably, fields [38]. In contrast, it follows from [15], [40], [42], [31] that Boolean algebras and linear orders are not universal. Harrison-Trainor, Melnikov, Miller, and Montalbán [22] have recently shown that all known proofs of universality share the same features. In particular, they are witnessed by Turing functionals that are also functors with some nice additional properties. Furthermore, they showed that there is an equivalent ${\mathcal{L}}_{{\omega }_1,\omega }^c$ definability-theoretic formulation of universality in all known cases. Their approach can be taken as the basic formal definition of (Turing) universality of a class; we however omit details.

The theorem below gives the first known example of a punctually universal class.

Theorem 1.2

The class of structures with only one binary function symbol is punctually universal.

We have not formally defined universality, let alone punctual universality. To state the result formally we need a few elementary definitions. We will give the formal definition of universality in the form of effective categories. This is essentially the same as [38], [21], except that we use p.r. functionals instead of computable functionals. The idea is that a class is universal if we can transform any structure into a structure in that class in a primitive recursive way, and that this transformation has an inverse transformation, and both transformations also respect isomorphisms. The formal definition will be given in the preliminaries section. We prove Theorem 1.2 in Section 3.

Since p.r. functionals are also Turing computable functionals, every punctually universal class is Turing universal as well. To show that a class is not universal, it is desirable to give a particular example of a primitive recursive property which cannot be realized within that class; in this way, the fact that the class is not universal in independent of the choice of a precise definition for universality.

A computable structure is computably categorical if it has exactly one (Turing) computable presentation up to (Turing) computable isomorphism [12], [16]. We say that a (punctual) structure is punctually categorical if between any two punctual copies of the structure there is a primitive recursive isomorphism with primitive recursive inverse [27]. In [27] Kalimullin, Melnikov and Ng constructed an example of a punctually categorical structure which is not computably categorical (and we conjecture that there are punctually categorical structures which are not $0^{(\alpha )}$-categorical). As we noted above, one should expect every punctually universal class to be Turing universal as well. In particular, a punctually universal class must also contain an example of this sort. On the other hand, the following theorem says that punctually categorical graphs are relatively simple.

Theorem 1.3

Let G be an undirected infinite graph. Then the following are equivalent:

• G is punctually categorical.

• G becomes a clique or an anti-clique (an independent set) after removing finitely many vertices $\overline{v}=v_0,\dots ,v_k$ with each $v_i$ being either adjacent to all $x\in (G-\overline{v})$ or not adjacent to all $x\in (G-\overline{v})$. Equivalently, G is automorphically trivial – there is a finite tuple $\overline{v}$ such that any permutation of G that fixes $\overline{v}$ pointwise is an automorphism.

In particular, every punctually categorical graph is computably categorical. So graphs are not universal for punctual computability for any reasonable notion of punctual universality. Interestingly, graphs are universal among automatic structures [30].

We remark that the proof of Theorem 1.3 involves several new proof techniques which seem like they should become basic to the area. They are not the usual priority arguments endemic in computability theory, but involve careful understanding of the relative speed of presentations of structures. Also involved is some kind of notion of “distance” within structures, something which is unexplored in classical structure theory in the non-metric setting.

We leave open:

Question 1.4

Is there a punctually universal relational class?

Likely our techniques will make the following sub-question accessible.

Question 1.5

Is there an algebraic description of punctually categorical directed graphs? Of punctually categorical structures with many binary relations? Of punctually categorical structures in a relational language?¹